Hypothesis Testing: Comparing Parameters (Means, Proportions, Incidence Rates) Between More Than 2 Populations With One Test

Lecture 11

Section A: (Hypothesis Testing) Comparing Means Between More Than Two Populations: Analysis of Variance (ANOVA)

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Learning Objectives

In this lecture section you will learn to interpret a p-value from a hypothesis test for (any) mean differences between more than two populations

The method for getting the p-value is called the Analysis of Variance (frequently called ANOVA)

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Example 1

Researchers were interested in the relationship between smoking and mid-expiratory flow (FEF), a measure of pulmonary health. The researchers recruited study subjects and classified them into one of six smoking categories1

Nonsmokers (NS) Passive smokers (PS) Non-inhaling smokers (NI) Light smokers (LS) Moderate smokers (MS) Heavy smokers (HS)

1 White, J.R., Froeb, H.F. Small-Airways Dysfunction in Non-Smokers Chronically Exposed to Tobacco Smoke, New England Journal of Medicine 302: 13 (1980)

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Example 1

To start, the researchers were interested in whether there were any statistically significant differences in pulmonary outcomes (FEV1, FEF, etc..) between the six underlying populations

One strategy would be to perform lots of two-sample t-tests (for each possible two-group comparison)

In this example, there would be 15 such comparisons ! NS to PS, NS to NI, and so on . . .

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Example 1

Analysis of Variance (One-Way ANOVA) is an extension of the two-sample (unpaired) t-test to compare means between more than two populations with one test

General idea behind ANOVA, comparing means for k-groups (k > 2):

Ho : µ1 = µ2 = …… µkHA : At least one mean different

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Example 1

Smoking and FEF From a pool of over 5,200 potential participants, a random

sample of 200 men and 200 women was drawn from each smoking group (except for the non-inhalers, where 50 men and 50 women were selected)

FEF measurements were taken on each of the subjects

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Example 1

Table 1

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Example 1

Data Summary, Males (FEF 25- 75%)

Group Mean FEF SD FEF

(L/s) (L/s) n

NS 3.78 0.79 200

PS 3.30 0.77 200

NI 3.32 0.86 50

LS 3.23 0.78 200

MS 2.73 0.81 200

HS 2.59 0.82 200

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Example 1

ANOVA comparing FEF values for males by smoking group

Ho: µNS=µPS=µNI=µLS=µMS=µHS

HA: At least two groups have different means

The p-value from ANOVA is very small: p < 0.01

Conclusion?

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Example 1

Overall summary of pulmonary comparisons, from abstract (based on follow-up two sample comparisons)

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Example 2

FEV1 and three medical centers 2

Data was collected on 60 patients with coronary artery disease at 3 difference medical centers (Johns Hopkins, Ranchos Los Amigos Medical Center, St. Louis University School of Medicine)

Purpose of study to investigate effects of carbon monoxide exposure on these patients

Prior to analyzing CO effects data, researchers wished to compared the respiratory health of these patients across the three medical centers

2 Pagano M, Gauvreau K. Principles of Biostatistics. Second Edition. Duxbury Press (2000).

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Boxplots

FEV1 Values by Center1.

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2.5

33.

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FEV1

(L/s

)

JH RLA SL

CHD Patients At 3 Medical CentersBoxplot of FEV1 Measurements

(n=21) (n=16) (n=23)

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Boxplots

ANOVA

Conclusions?

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Example 2

ANOVA approach, conceptually

Assume the null hypothesis is true Compute a measure of discrepancy between what was observed

in the sample compared to what is expected under the null (in ANOVA this is called a “F-statistic”

Compared this measure of discrepancy (F-statistic) to the distribution of such measures because of random sampling variability, when the null is true

Convert to a p-value

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Example 2

For this example, the F-statistic is 3.12. To get a p-value, this observed value is compared to a F-distribution with 2 numerator, and 57 denominator degrees of freedom

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Example 2

For this example, the F-statistic is 3.12. To get a p-value, this observed value is compared to a F-distribution with 2 numerator, and 57 denominator degrees of freedom

Example 3: Academic Physician Salaries3

From abstract

3 Jagsi R, et al. Gender Differences in the Salaries of Physician Researchers. Journal of the American Medical Association (2012); 307(22); 2410-2417.

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Example

Article Table

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Example 4: Academic Physician Salaries

Article Table

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Summary

Section B: (Hypothesis Testing) Comparing Proportions Between More Than Two Populations: Chi-Square Tests

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Learning Objectives

In this lecture set you will learn to interpret a p-value for a hypothesis test comparing proportions between more than two populations

The method for getting the p-value is an extension of the chi square test shown in lecture 10, and is also called a chi square test

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Example 1

Health care indicators by immigrant status1

1 Huang, Z., et al. Health Status and Health Service Access and Use Among Children in U.S. Immigrant Families, American Journal of Public Health 96: 4 (2006)

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Example 1

Example: health care indicators by immigrant status1

1 Huang, Z., et al. Health Status and Health Service Access and Use Among Children in U.S. Immigrant Families, American Journal of Public Health 96: 4 (2006)

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Example 1

Zoom in

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Example 1

Chi Square Approach

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Example 1

Zoom in

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Example 2

Out of pocket spending, medication adherence: dialysis patients, 12 countries2

2 Hirth R, et al. Out-Of-Pocket Spending And Medication Adherence Among Dialysis Patients In Twelve Countries. Health Affairs (2008). 27 (1).

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Example 2

Characteristics of dialysis patients, 12 countries (six shown)

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Example 2

Percentage of minorities, compared across the 12 countries (only 6 countries shown)

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Example 2

Percentage of minorities, compared across the 12 countries (only 6 countries shown)

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Example 2

Distribution of income categories

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Example 2

Distribution of income categories

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Example 2

ANOVA as well (previously lecture section infiltration!)

Example 3: Academic Physician Salaries3

From abstract

3 Jagsi R, et al. Gender Differences in the Salaries of Physician Researchers. Journal of the American Medical Association (2012); 307(22); 2410-2417.

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Example 3: Academic Physician Salaries3

Table: characteristics of physicians by sex

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Example 3: Academic Physician Salaries3

Table: characteristics of physicians by sex

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Example 3: Academic Physician Salaries3

Table: characteristics of physicians by sex

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Summary

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Section C: (Hypothesis Testing) Comparing Survival Curves Between More Than Two Populations: Log-rank Tests

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Learning Objectives

In this lecture set you will learn interpret a p-value for a hypothesis test comparing survival curves (and hence incidence rates) between more than two populations

The method for getting the p-value is an extension of the log rank test shown in lecture 10, and is also called a log rank test

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Maternal Vitamin Supplementation and Infant Mortality

1 Katz J, West K et al. Maternal low-dose vitamin A or β-carotene supplementation has no effect on fetal loss and early infant mortality: a randomized cluster trial in Nepal. American Journal of Clinical Nutrition (2000) Vol. 71, No. 6, 1570-1576.

Example 11

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05.1deaths/day 00039.0deaths/day 00041.0

ˆˆˆ placebo

vitAvitA RI

RIRIR

00.1deaths/day 00039.0deaths/day 00039.0

ˆˆˆ placebo

BCBC RI

RIRIR

Maternal Vitamin Supplementation and Infant Mortality

Example 1

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0.90

0.95

1.00

Prop

ortio

n St

ill Al

ive

0 50 100 150 200Follow-Up Time Since Enrollment in Study (days)

treat = Beta C treat = Placebotreat = Vit A

Kaplan-Meier survival estimates

Maternal Vitamin Supplementation and Infant Mortality

Example 1

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0.00

0.05

0.10

Prop

ortio

n Th

at H

ad D

ied

0 50 100 150 200Follow-Up Time Since Enrollment in Study (days)

treat = Beta C treat = Placebotreat = Vit A

Kaplan-Meier failure estimates

Log rank test

Interpretation?

Example 1

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Examples 2

Return to Work Following Injury: The Role of Economic, Social, and Job-Related Factors2

“The main dependent variable in the analysis is the time (in days) from injury to the first time the study patient returned to work. Kaplan-Meier estimates of the cumulative proportion of patients returning to work were computed. These estimates take into account how long patients were followed as well as when they returned to work. A log-rank test was used to test the association between the cumulative probability of RTW and each of the risk factors considered one at a time”

2 MacKenzie E, et al Return to Work Following Injury:The Role of Economic, Social, and Job-Related Factors America Journal of Public Health 88; 11 (1998)

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Example 2

Kaplan Meier estimates of )(ˆ1 tS

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Example 3

Post partum ART therapy for children born to HIV positive women3

3 Nielson-Saines K, et al. Three Postpartum Antiretroviral Regimens to Prevent Intrapartum HIV Infection. New England Journal of Medicine (2012). 366 (25).

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Example 3

Post partum ART therapy for children born to HIV positive women

“Kaplan–Meier curves for intrapartum transmission differed significantly (P = 0.03 for the overall comparison). Transmission rates were highest in the zidovudine-alone group (3.4% at 4 to 6 weeks vs. 1.6% in the two-drug group and 1.4% in the three-drug group; 4.8% at 3 months vs. 2.2% in the two-drug group and 2.4% in the three-drug group).”

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Example 3

Post partum ART therapy for children born to HIV positive women

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Summary